Taking into account the increasing availability of computational power at an affordable cost, two-dimensional through-flow calculation methods are gaining more and more attention, given the fact that the required time for convergence is continuously reducing. Consequently, several application fields (i.e. whole engine performance simulation), that were traditionally dominated by simpler and faster zero-dimensional or one-dimensional methods, purely because of computational power restrictions, gradually move towards two-dimensional analyses. These tend to offer more information about the flow-field at a greater accuracy. The Radial Equilibrium Equation (REE), in its either simple or full version, has been the basis of several two-dimensional and quasi-three-dimensional through-flow techniques that are being used for the flow analysis within ducts, compressors and turbines. The aim of this paper is to provoke a thorough discussion on the actual solution of the full REE for the determination of the meridional velocity profile. More precisely, this manuscript discusses in detail the implications on the solution of the full REE when the blade lean angle related terms are included in the equation. This issue has only been superficially addressed in the existing literature up to this stage. The expressions for radial equilibrium addressed in the context of this paper, mainly consist the basis of a particular streamline curvature code (2D SLC Compressor Software), developed as a performance investigation and design tool of axial flow compressors. This code has been through a number of ‘improvement cycles’ over its several years of existence. One such cycle included the elaborate study of several final versions of the full REE, in order to reassure a stable and fast convergence for the final solution, while maintaining the highest possible level of accuracy. Firstly, this manuscript presents the final version of the full REE, commenting on each individual term in the equation, as well as on the various assumptions made during its derivation process. The two different solutions of the equation are given for zero and non-zero blade lean angle values. Moreover, the implications of the solution of the non-zero blade lean angle equation on the stability, convergence time and accuracy of the final results are pointed out. Finally, some conclusions are expressed as far as the effects of the blade lean angle on a compressor blade row performance and the actual applicability of the two forms of the REE are concerned. These conclusions were drawn from personal experience applying the equations but also from an extensive literature review conducted.
Inverse Design Tool for Ion Optical Devices using the Adjoint Variable Method